Let \(F_q(z_i,\theta ) = C_q(z_i)\Vert \theta - \theta ^\dagger \Vert  + \Vert \partial _\theta q(z_i,\theta ^\dagger )\Vert \)  be the enveloppe function which satisfies: \(\Vert \partial _\theta q(z_i,\theta )\Vert  \le F_q(z_i,\theta )\)  for all \(\theta \in \Theta \) . We have \(F_q(z_i,\theta ) \le F_q(z_i) = C_q(z_i)\text{diam}(\Theta ) + \Vert \partial _\theta q(z_i,\theta ^\dagger )\Vert \) , which does not depend on \(\theta \) . Here \(\text{diam}(\Theta ) = \sup _{\theta _1,\theta _2 \in \Theta }\Vert \theta _1-\theta _2\Vert \) , by compactness it is finite. The main idea is to apply Theorem 2.14.5 in [1]}, under the distribution \(\mathbb {P}^\star \) , for a \(p \ge 2\)  such that Assumption REF  holds:
\(&\left[ \mathbb {E}^\star \left( \sup _{\theta \in \Theta } \sqrt{m}\Vert G_m^{(b)}(\theta ) - G_n(\theta )\Vert ^p \right) \right]^{1/p}\\&\lesssim \mathbb {E}^\star \left( \sup _{\theta \in \Theta } \sqrt{m}\Vert G_m^{(b)}(\theta ) - G_n(\theta )\Vert  \right) + m^{1/p-1/2}\left[ \frac{1}{n} \sum _{i=1}^n |F_q(z_i)|^p \right]^{1/p},\) 
